kumalsamir89
Answered

The population of a town increases every year by [tex]$10\%$[/tex]. At the end of two years, the total population of the town was 30,000. If it includes 5,800 people who were added by migration,

(a) If the initial population is [tex]$P$[/tex] and the population after [tex]$T$[/tex] years is [tex]$P_T$[/tex], write the formula to calculate [tex]$P_T$[/tex].
[tex]\[ P_T = P \left(1 + \frac{R}{100}\right)^T \][/tex]

(b) Write the population after 2 years.
[tex]\[ P_T = 30,000 \][/tex]

(c) Find the population before 2 years.
[tex]\[ P = \frac{P_T - 5800}{\left(1 + \frac{R}{100}\right)^2} \][/tex]
[tex]\[ P = \frac{30,000 - 5,800}{(1.10)^2} \][/tex]
[tex]\[ P \approx 20,000 \][/tex]

(Note: Removed irrelevant part about "simple population growth in four years" as it doesn't fit the context of the task given.)



Answer :

GinnyAnswer
(a) To find the initial population [tex]\( P \)[/tex] given the population after [tex]\( T \)[/tex] years, denoted as [tex]\( P_T \)[/tex], and the growth rate [tex]\( R \)[/tex], the formula used is:

[tex]\[ P = \frac{P_T}{(1 + \frac{R}{100})^T} \][/tex]

This formula accounts for compound growth over [tex]\( T \)[/tex] years.

(b) The population after 2 years ([tex]\( P_T \)[/tex]) is given as 30000. This includes 5800 people who migrated into the town.

So, the population due to natural growth, excluding migration, is:

[tex]\[ \text{Population excluding migration} = 30000 - 5800 = 24200 \][/tex]

(c) Now, to find the initial population [tex]\( P \)[/tex] before 2 years, we rearrange the formula:

[tex]\[ P = \frac{24200}{(1 + \frac{10}{100})^2} \][/tex]

Calculating this using the given data:

[tex]\[ P = \frac{24200}{(1 + 0.10)^2} \][/tex]
[tex]\[ P = \frac{24200}{1.21} \approx 20000 \][/tex]

So, the initial population before 2 years was approximately 20000.